# If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.

If $$Y$$ is a Hausdorff space and $$X$$ which is a subspace of that space is limit point compact space . Then $$X$$ is closed.

Can anyone give me a trivial counter example?

• Can you define the term 'limit point compact'? – Kavi Rama Murthy Dec 6 '18 at 5:51
• If $X$ is a limit point compact space then every infinite set has a limit point. E.g - $\mathbb R$ is not limit point compact space where so is any of it's closed interval.@KaviRamaMurthy – cmi Dec 6 '18 at 5:59

## 1 Answer

Let $$Y$$ be $$\omega_1 + 1$$ and $$X$$ is $$\omega_1$$, in the order topology. Here $$\omega_1$$ is the first uncountable ordinal, and $$Y$$ is its successor (one extra point).

Another example: let $$Y$$ be $$[0,1]^\mathbb{R}$$ in the product topology, and $$X$$ the limit point compact subset of all elements that are $$0$$ except for at most countably many coordinates. ( A $$\Sigma$$-product, this is called). This $$X$$ is dense and not closed in $$Y$$.

• what is first countable ordinal? Is there no simple counter example?@Henno Brandsma – cmi Dec 6 '18 at 6:20
• @cmi Munkres calls it $W$. Look it up – Henno Brandsma Dec 6 '18 at 7:02